# homotopy
## English
### Etymology

From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.

### Noun

homotopy (countable and uncountable, plural homotopies)
* (topology) A continuous deformation of one continuous function or map to another.
    The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.
** 1998, Paul Sellick, “Space Exponents for Loop Spaces of Spheres”, in William G. Dwyer, editor, Stable and Unstable Homotopy, American Mathematical Society, page 279:An integer M is called an exponent for the torsion of an abelian group G if M * (torsion of G) = 0. We say that M is a homotopy exponent for a space X if M is an exponent for π$_k$ (X) for all k.
** 2001, F. R. Cohen, S. Gitler, “Loop-spaces of configuration spaces, braid-like groups, and knots”, in Jaume Aguadé, Carles Broto, Carles Casacuberta, editors, Cohomological Methods in Homotopy Theory, Springer (Birkhäuser), page 63:A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below.
** 2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,
    In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X.
* (uncountable) The relationship between two continuous functions where homotopy from one to the other is evident.
* (informal) Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes).
* (topology) A theory associating a system of groups with each topological space.
* (topology) A system of groups associated with a topological space.

#### Usage notes

* Formally, there are two alternative formulations:
** Given topological spaces $X,Y$ and continuous maps $f,g:X\rightarrow Y$
*** A continuous map $H:[0,1]\times X\rightarrow Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ $\forall x\in X$.
*** A family of continuous maps $h_{t}:X\rightarrow Y,t\in [0,1]$ such that $h_{0}=f,$ $h_{1}=g$ and the map $(x,t)\mapsto h_{t}$ is continuous from $X$ to $Y$. (Note that it is not sufficient to require that each map $h_{t}(x)$ be continuous.)
** Replacing the unit interval $[0,1]$ with the affine line A¹ leads to A¹ homotopy theory.
* The adjective homotopic is used specifically in the sense, with respect to two functions, of "having the relationship of being in homotopy".
** Being homotopic is an equivalence relation on the class of all continuous functions between given topological spaces. An equivalence class of such a relation is called a homotopy class.

#### Hyponyms

* (continuous deformation): isotopy, regular homotopy

#### Derived terms

* cohomotopy
* homotopic
* homotopy category
* homotopy class
* homotopy equivalence
* homotopy equivalent
* homotopy fiber, homotopy fibre
* homotopy group
* homotopy invariant
* homotopy relative to a subspace
* homotopy theory
* homotopy type
* linear homotopy
* regular homotopy
* straight-line homotopy

#### Related terms

* homotopic

#### Translations
### See also

* homeotopy group
* homology

### References

* ↑ Earliest Known Uses of Some of the Words of Mathematics (H)
* ↑ Homotopy on Wikipedia.Wikipedia

### Further reading

* Homotopy group on Wikipedia.Wikipedia
* Fundamental group on Wikipedia.Wikipedia
* Homotopy theory on Wikipedia.Wikipedia
* A¹ homotopy theory on Wikipedia.Wikipedia
* Homeotopy on Wikipedia.Wikipedia
* Fiber-homotopy equivalence on Wikipedia.Wikipedia
* Poincaré conjecture on Wikipedia.Wikipedia
* Homotopy on Encyclopedia of Mathematics